Section 3.4 Cross Product

u × v = (u2v3 − u3v2, u3v1 − u1v3, u1v2 − u2v1) THE CROSS PRODUCT If u = (u1, u2, u3) and v = (v1, v2, v3) are vectors in 3-space, then the cross product or u × v is defined by u × v = (u2v3 − u3v2, u3v1 − u1v3, u1v2 − u2v1) or in determinant notation,

STANDARD UNIT VECTORS The standard unit vectors in 3-space are the vectors i = (1, 0, 0), j =(0, 1, 0), k = (0, 0, 1). Each of these vectors have length 1 unit and lie on the coordinate axes.

COMMENTS ON STANDARD UNIT VECTORS 1. Every vector in 3-space can be expressed in terms of the standard unit vectors: v = (v1, v2, v3) = v1i + v2j + v3k 2.

DETERMINANT FORM OF THE CROSS PRODUCT If u = (u1, u2, u3) and v = (v1, v2, v3), then

RELATIONSHIPS INVOLVING CROSS PRODUCT AND DOT PRODUCT Theorem 3.4.1: If u, v, and w are vectors in 3-space, then (a) u ∙ (u × v) = 0 (u × v is orthogonal to u) (b) v ∙ (u × v) = 0 (u × v is orthogonal to v) (c) ||u × v||2 = ||u||2 ||v||2 − (u ∙ v)2 (Lagrange’s identity) (d) u × (v × w) = (u ∙ w) v − (u ∙ v) w (e) (u × v) × w = (u ∙ w) v − (v ∙ w) u

PROPERTIES OF THE CROSS PRODUCT Theorem 3.4.2: If u, v, and w are any vectors in 3-space and k is a scalar, then (a) u × v = −(v × u) (b) u × (v + w) = (u × v) + (u × w) (c) (u + v) × w = (u × w) + (v × w) (d) k(u × w) = (ku) × w = u × (kw) (e) u × 0 = 0 × u = 0 (f) u × u = 0

AREA OF A PARALLELOGRAM Theorem 3.4.3: If u and v are vectors in 3-space, then || u × v || is equal to the area of the parallelogram determined by u and v.

THE SCALAR TRIPLE PRODUCT If u, v and w are vectors in 3-space, then the scalar triple product of u, v, and w is defined by u · (v × w).

COMMENTS ON THE TRIPLE SCALAR PRODUCT 1. If u = (u1, u2, u3), v = (v1, v2, v3) and w = (w1, w2, w3), the triple scalar product can be calculated by the formula 2. u · (v × w) = w · (u × v) = v · (w × u)

GEOMETRIC INTERPRETATION OF DETERMINANTS Theorem 3.4.4: Part (a) The area of the parallelogram in 2-space determined by the vectors u = (u1, u2) and v = (v1, v2) is given by

GEO. INTERPRETATION OF DET. (CONCLUDED) Theorem 3.4.4: Part (b) The volume of the parallelepiped in 3-space determined by the vectors u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3) is given by

VECTORS IN THE SAME PLANE Theorem 3.4.5: If the vectors u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3) have the same initial point, then they lie in the same plane if and only if